The Coordinate Method – Explained, with a Shortcut

Per the CA BPESLG’s test plan for the civil surveying exam, test takers need to know how to perform “trigonometric relationship to determine the area of a polygon” and “procedures for calculating area” when it comes to traverses. If you’re taking a course or self-studying, then you’ve probably heard of these three methods for calculating the enclosed area of a traverse/polygon:

In my opinion, the easiest way to calculate the enclosed area of a traverse is the coordinate method, since it is simply a plug-n-chug problem (so is the DMD method, but it has more rules to follow) if you’re given coordinates or can easily resolve the polygon’s vertices into a coordinate system.

So, let’s take a look at the coordinate method formula for a 4-sided traverse with points A, B, C, and D:

What the heck? At first glance, this doesn’t seem like a simple plug-n-chug formula. Also, why is YA reintroduced at the end, and why is YD in the beginning term? The formula looks pretty daunting, but here’s a better way to visualize what’s going on in that nasty numerator (this figure was inspired from Chapter 6.8 from Cuomo, 2nd edition)

The products of the solid lines are positive, while the products of the dashed lines are negative (compare this to the previous formula to confirm). From this diagram above, you can see pretty clearly why the coordinate method is also called the “criss-cross” method, since you must go around the traverse, coordinate-by-coordinate, and multiply by the coordinates before and after by the one you are currently working on. You’ll notice that I have highlighted (XA, YA) in red and (XD, YD) in blue to illustrate this point. If you were finishing the calculation, you would have to multiply Point D’s x-coordinate with the y-coordinates from Point C, and Point A (the beginning point).

If that explanation confused you, don’t worry, there’s a a shortcut to this madness and no need to recall any formula!

(Somewhat of a) Shortcut to using the Coordinate Method

The best way to calculate an area using the coordinate method is by setting up a table for your criss-cross calculations. Instead of giving you a list of steps, I’ll show you an example:

Question: What is the area of the 4-sided traverse below?

Solution:

Step 1.) Resolve the traverse or polygon into (x,y) or (N, E) coordinates: You’ll see that I was nice and gave you the coordinates already, but if you’re not given any coordinates and it’s too complicated to resolve the polygon into triangles and rectangles, make up your own coordinates for each vertex (it’s nice to label one of the points strategically as (0,0) to ease calculations).

Step 2.) Create a two-row table, with as many columns as there are unique points , plus one more (in this case 4 +1 = 5). The top row is for x-coordinates, bottom row is for y-coordinates. It really doesn’t matter which row you use, or if it’s Northing and Eastings, but what does matter is that you must write the coordinates in order, going clockwise or counterclockwise, starting at any vertex. Also, you must enter the same coordinates you started with in the last column (in this case point A was entered in the first and last columns).

Step 3.) Beginning with the first column and moving right, do a “criss-cross” calculation for the top row. In other words, multiply the first column/first row by the second column/second row, and write the answer underneath (shorthand notation for big numbers). Continue this for each column, moving right.

Step 4.) Do your “criss-cross” with the bottom row, writing the answer on top:

Step 5.)Calculate the sum of the top and the bottom. Be sure to include negative signs in your summation when you have negative coordinate(s).

Step 6.) Take the absolute value of the difference between the sum of top and sum of bottom. Divide this number by 2, and that’s your answer.

The above process worked well for me on the exam, and it’s pretty fast once you get the hang of it. Hell, as long asyou set up the table, you could probably do all these calculations in one or two steps on your calculator. The table just helps you visualize the sequence.

If you liked this “shortcut”, here’s what I would recommend writing in your references to help you remember the process:

Four a four-sided (for simplicity) traverse/polygon, with points A, B, C, and D:

If given latitudes and departures, converting to coordinates is just one extra step. You would need to assign an arbitrary coordinate, preferably (0, 0) to a beginning point of your choice and calculate the coordinates of the other points relative to that beginning point.

For example, here are the latitudes and departures for the four-sided polygon problem in this post:

If you choose point A as your beginning point and assign it as (0, 0), then:

Notice that the traverse closes back on itself perfectly at point A. This is a good self check for your calculations.

After you’ve setup that table, you can use the coordinate method shortcut and you’ll end up with the same area I got in the above post (614295.5 ft2 which rounds to 610,000 ft2). Hope this helps!

Note: There is another method called the Double-Meridian-Distance method (DMD) which is more suited for calculating areas when given only the latitude and departure, but it’s also a tedious method and I think the coordinate method is more straightforward. I’ll write a blog post about the DMD method in the future.

Yes you’re correct. This method works with any # sided polygon. If you are dealing with more than four coordinates you would continue the pattern and make sure to return to the first term in the last column as shown above.

I took a look at the FE Handbook and I see they have a formula for “Area by Coordinates”:

It does return to YA in the last term which is correct since your calculations have to “close” at the beginning point. The last column which repeats the first term accounts for this. However, to me the FE formula is too hard to read. If you follow the setup in my post it is exactly the same as what the formula is trying to say :).

Points X(m) Y(m)
A 0 0
B 50 200
C 150 50
it is required to divide the area into two equal parts by a line passing through point A, calculate the coordinates of the other end of the dividing line. (its says by coordinates method only)

I would say to assume there is a point “R” with unknown X & Y coordinates between B & C since that’s the only place it could be. Now, consider this figure below:

Since we want point R to give you 1/2 of the area (which I believe is already given?), you can setup the coordinate method table for the triangle above. Next, you should setup the equation to solve for area, but make sure you are solving for half of the area given since point R is the dividing point. By doing this you will end up with two unknowns: Rx and Ry.

BUT, You still need another equation to relate Rx and Ry to each other. Luckily since Point R is on a line between points B and C, you can interpolate between the two points and up with something like this:

Now you’ll have two equations with two unknowns and should be able to solve for Rx and Ry.